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------------------------------------------------ Find all the roots of [tex]$f(x) = x^4 - 2x^3 - 48x^2$[/tex]. Show all work for full credit.

To find all the roots of the polynomial function [tex]$ f(x) = x^4 - 2x^3 - 48x^2 $[/tex], we'll go through some algebraic steps.

### Step 1: Factor Out the Common Term

The given polynomial has a common factor of [tex]$ x^2 $[/tex] in each term. We'll factor out [tex]$ x^2 $[/tex]:

[tex]\[
f(x) = x^2(x^2 - 2x - 48)
\][/tex]

### Step 2: Solve the Simple Root

From the factored form, we have:

[tex]\[
x^2 = 0
\][/tex]

Solving [tex]$ x^2 = 0 $[/tex] gives:

[tex]\[
x = 0
\][/tex]

This is one of the roots with multiplicity 2.

### Step 3: Solve the Quadratic Expression

Next, we need to solve the quadratic equation:

[tex]\[
x^2 - 2x - 48 = 0
\][/tex]

We can use the quadratic formula, which is expressed as:

[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

For the quadratic equation [tex]$ x^2 - 2x - 48 = 0 $[/tex], the coefficients are [tex]$ a = 1 $[/tex], [tex]$ b = -2 $[/tex], and [tex]$ c = -48 $[/tex].

Plugging these values into the quadratic formula:

[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1}
\][/tex]

[tex]\[
x = \frac{2 \pm \sqrt{4 + 192}}{2}
\][/tex]

[tex]\[
x = \frac{2 \pm \sqrt{196}}{2}
\][/tex]

[tex]\[
x = \frac{2 \pm 14}{2}
\][/tex]

This results in two potential solutions:

1. [tex]$ x = \frac{2 + 14}{2} = 8 $[/tex]
2. [tex]$ x = \frac{2 - 14}{2} = -6 $[/tex]

### Step 4: Compile the Roots

So, the roots of the polynomial [tex]$ f(x) = x^4 - 2x^3 - 48x^2 $[/tex] are:

- [tex]$ x = 0 $[/tex] with multiplicity 2
- [tex]$ x = 8 $[/tex]
- [tex]$ x = -6 $[/tex]

These three are all the roots of the polynomial given.